As defined in section 2.8 of [1]}, a d-critical structure on an algebraic stack \(X\)  is equivalent to the data of d-critical structure \(y^\star (s)\)  on the scheme \(Y\)  for each smooth 1-morphism \(y:Y\rightarrow X\)  such that for each 2-morphism \((\phi ,\eta )\)  between the 1-morphism \(y:Y\rightarrow X\)  and \(z:Z\rightarrow X\) , \(\phi ^\star z^\star (s)=y^\star (s)\) . In particular, [2]} shows that a (-1)-shifted symplectic algebraic stack has a canonical d-critical structure. On a d-critical stack \(X\) , Joyce built its canonical sheaf \(K_X\) , which is the determinant of the tangent complex if \(X\)  is the classical truncation of a (-1)-shifted symplectic stack. An orientation \(K_X^{1/2}\)  is then a square root of \(K_X\) , and [1]} shows that an orientation \(K_X^{1/2}\)  determines an orientation of \((Y,y^\star (s))\)  for each scheme \(Y\)  and 1-morphism \(y:Y\rightarrow X\) . In particular, a smooth 2-morphism \((\phi ,\eta )\)  of relative dimension \(d\)  gives then a smooth morphism of oriented critical scheme \(\phi :(X,y^\star (s),K_Y^{1/2})\rightarrow (Y,z^\star (s);K_Z^{1/2})\)  of relative dimension \(d\) .
